The problems of asymptotic controllability and feedback stabilization to a point or to a set
of control systems
that are nonlinear (and for which linearization fails),
and their relationships with the existence of control Lyapunov functions have been central topics in control theory since the 1980s. It is now well-known that a smooth control Lyapunov function, which guarantees the asymptotic controllability of the system, may not exist and a continuous stabilizing feedback fails in general to exist either (see [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}). Under suitable assumptions on the dynamics function \(f\)  and on the control set \(U\) , these problems were solved in [8]} by the introduction of nonsmooth control Lyapunov functions, discontinuous feedback laws \(K=K(x)\) , and a ‘‘sample and hold" solution concept, similar to that used in differential games in [9]}.
